← internal-hom-conway-addition

main.tex

\documentclass[11pt,reqno]{amsart}

\usepackage{amsmath,amssymb,amsthm}
\usepackage{mathtools}
\usepackage[margin=1.15in]{geometry}
\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref}
\usepackage{microtype}

\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{question}[theorem]{Question}

\newcommand{\catGame}{\mathbf{Game}}
\newcommand{\Set}{\mathbf{Set}}
\newcommand{\Pfin}{\mathcal{P}_{\mathrm{fin}}}
\newcommand{\HF}{\mathbb{H}}
\newcommand{\unit}{\mathbf{1}}
\newcommand{\ihom}[2]{[#1,#2]_{+}}
\newcommand{\ev}{\operatorname{ev}}
\newcommand{\desc}{\operatorname{desc}}
\DeclareMathOperator{\im}{im}

\begin{document}

\title[Internal hom for Conway addition]{An internal hom for Conway addition}

\author{Ryuya Hora}
\thanks{This is a private research memorandum, drafted with AI collaborators
(Claude, Codex) on the basis of the author's Kobin research vault. It is
circulated privately and is not intended for public distribution; it has not
been refereed, and its claims should be read with that in mind.}

\date{August 26, 2026}

\subjclass[2020]{91A46, 18M05, 18C50}
\keywords{combinatorial game, recursive coalgebra, Conway addition,
monoidal closed category, internal hom}

\begin{abstract}
In \emph{Games as recursive coalgebras} the category $\catGame$ of impartial
combinatorial games is shown to be symmetric monoidal closed for the Conway
addition, but only by an abstract adjoint functor argument; Question~5.7 of
that paper (Problem~5.0.5 of the author's problem list) asks what the internal
hom actually is. We answer this. For games $G,H$ we exhibit $\ihom{G}{H}$ as
the largest subgame $A(G,H)$ of the cofree game on the set $\Set(UG,UH)$ on
which evaluation of labels is a game morphism. Concretely, a position of
$\ihom{G}{H}$ is a finite well-founded tree whose nodes are labelled by
functions $UG\to UH$, subject to a hereditary option equation, and a move is
the passage to a child. We prove that evaluation is a game morphism, that
$A(G,H)$ is maximal with this property, that currying is a bijection natural in
all three variables, and we compute several examples. In particular
$\ihom{\ast 1}{\ast n}\cong\unit$ for every $n\ge 1$ and
$\ihom{\ast 2}{\ast 1}=\emptyset$, so this internal hom is emphatically not the
$-G+H$ of Joyal's compact closed category of games and strategies. The
relation between the two remains open.
\end{abstract}

\maketitle

\section{Introduction}

Impartial combinatorial games can be organised into a category $\catGame$ whose
objects are games in the sense of Conway --- a set of positions together with a
finitely branching, well-founded option relation --- and whose morphisms are
maps that both preserve and lift options. Under the identification of games
with recursive $\Pfin$-coalgebras \cite[Theorem~3.14]{Hora}, $\catGame$ is a
locally finitely presentable category \cite[Theorem~B.9]{Hora}, comonadic over
$\Set$ \cite[Theorem~3.15]{Hora}, and the Conway addition makes it symmetric
monoidal closed \cite[Proposition~4.2]{Hora}.

The proof of closedness given there is indirect. By
\cite[Corollary~B.11]{Hora}, \emph{every} monoidal structure on $\catGame$
lifting the cartesian structure of $\Set$ is closed, because each
$G+(-)$ is cocontinuous and the adjoint functor theorem for locally
presentable categories applies. This produces a right adjoint but says
nothing about what its positions and moves are. Accordingly:

\begin{question}[{\cite[Question~5.7]{Hora}}; Problem~5.0.5 of \cite{HoraProblems}]
\label{q:main}
What is the internal hom with respect to the Conway addition?
\end{question}

This note answers Question~\ref{q:main} by an explicit construction. Write
$UG$ for the set of positions of a game $G$ and $\theta_G$ for its option map.
Let $S=\Set(UG,UH)$ and let $R(S)$ be the cofree game on $S$, whose positions
are the $S$-labelled hereditarily finite sets \cite[Proposition~B.14]{Hora}.
Call a position $u$ of $R(S)$ \emph{compatible} if for every descendant $w$ of
$u$ (including $u$ itself) and every $g\in UG$,
\stepcounter{equation}\begin{equation}\tag{E}\label{eq:E}
  \theta_H\bigl(a_w(g)\bigr)
  \;=\;\bigl\{\,a_w(g')\;\big|\;g'\in\theta_G(g)\,\bigr\}
  \;\cup\;\bigl\{\,a_v(g)\;\big|\;v\in\theta_{R}(w)\,\bigr\},
\end{equation}
where $a_w\colon UG\to UH$ is the label of $w$. Let $A(G,H)\subseteq R(S)$ be
the set of compatible positions.

\begin{theorem}[Theorem~\ref{thm:main} below]
$A(G,H)$ is a subgame of $R(S)$, the label evaluation
$\ev(g,u)=a_u(g)$ is a game morphism $G+A(G,H)\to H$, and for every game $K$
currying is a bijection
\[
  \catGame(G+K,\,H)\;\xrightarrow{\ \cong\ }\;\catGame\bigl(K,\,A(G,H)\bigr),
\]
natural in $G$, $K$ and $H$. Hence $A(G,H)$ is the internal hom
$\ihom{G}{H}$ for the Conway addition.
\end{theorem}

Equation~\eqref{eq:E} has a direct game-theoretic reading, recorded in
Section~\ref{sec:reading}: a position of $\ihom{G}{H}$ is a \emph{plan},
that is, a map of option graphs $a\colon G\to H$, together with the finite set
of plans reachable in one move, such that the options of $a(g)$ in $H$ are
\emph{exactly} accounted for --- with no repetition of accounting and no
omission --- by the moves of $G$ at $g$ and by the moves of the plan itself.
Playing in $\ihom{G}{H}$ is revising the plan.

Section~\ref{sec:examples} computes examples. The unit law
$\ihom{\unit}{H}\cong H$ holds; $\ihom{G}{\unit}$ is $\unit$ or $\emptyset$
according to whether $G$ is discrete; and for the one-heap Nim games
$\ast n$ we find $\ihom{\ast 1}{\ast n}\cong\unit$ for all $n\ge1$ and
$\ihom{\ast 2}{\ast 1}=\emptyset$. These already show that the present
internal hom is not a categorification of Nim arithmetic and, in particular,
does not agree with the $-G+H$ of the classical strategy categories of
Joyal \cite{Joyal} and Honsell--Lenisa--Redamalla \cite{HLR}, where the
morphism structure is by winning strategies rather than by exact coalgebra
maps.

\medskip
\noindent\textbf{What is and is not claimed.}
Everything stated as a lemma, proposition or theorem below is proved here
from the definitions; the only external inputs are the definitions and the
description of the cofree game from \cite{Hora}, and even the universal
property of the latter is reproved in Lemma~\ref{lem:cofree} so that the
argument is self-contained. What is \emph{not} claimed: (i) any relation
between $\ihom{G}{H}$ and player-switching duality, negation, or copycat
strategies --- the present setting does not contain the data these require,
and Section~\ref{sec:open} records what remains open; (ii) that the same
maximal-compatible-subobject construction works for an arbitrary bifunctor
lifting the cartesian product of $\Set$, which would answer
\cite[Questions~5.5, 5.6]{Hora}; (iii) novelty. Regarding (iii): we have not
been able to locate the construction below in the literature, but carving a
mapping object out of a cofree object by imposing a structure-preservation
condition is a general and old technique, and a negative search is not a
proof of novelty.

\section{Preliminaries}\label{sec:prelim}

We follow \cite{Hora}. All the material in this section is from there,
except Lemma~\ref{lem:cofree}, whose statement is
\cite[Proposition~B.14]{Hora} but whose proof we give in the form we need.

\begin{definition}[{\cite[Definition~2.1]{Hora}}]
A \emph{game} $G$ is a set $UG$ together with a map
$\theta_G\colon UG\to\Pfin(UG)$ (equivalently a relation $\to_G$ with
$\theta_G(g)=\{g'\mid g\to_G g'\}$) such that every option set is finite and
there is no infinite path $g_0\to_G g_1\to_G\cdots$.
\end{definition}

Finite branching plus the absence of infinite paths gives, by K\"onig's lemma,
a \emph{rank} $\rho_G(g)\in\mathbb{N}$: the length of the longest path starting
at $g$. We use induction on $\rho_G$ freely.

\begin{definition}[{\cite[Definition~3.12, Notation~3.13]{Hora}}]
A \emph{game morphism} $f\colon G\to H$ is a function $f\colon UG\to UH$ that
is a graph morphism and lifts paths; equivalently, one for which
\stepcounter{equation}\begin{equation}\tag{M}\label{eq:M}
  f\bigl[\theta_G(g)\bigr]=\theta_H\bigl(f(g)\bigr)\qquad(g\in UG).
\end{equation}
Games and game morphisms form the category $\catGame$, and
$U\colon\catGame\to\Set$ is the forgetful functor.
\end{definition}

We stress that \eqref{eq:M} is an \emph{equality} of sets, not an inclusion:
$\catGame$ is the category of recursive $\Pfin$-coalgebras and coalgebra
homomorphisms \cite[Theorem~3.14]{Hora}, and this exactness is what makes
the computations of Section~\ref{sec:examples} so rigid.

\begin{definition}[{\cite[Definition~2.10]{Hora}}]
The \emph{Conway addition} $G+K$ has $U(G+K)=UG\times UK$ and
\[
  \theta_{G+K}(g,k)=\bigl(\theta_G(g)\times\{k\}\bigr)\cup
  \bigl(\{g\}\times\theta_K(k)\bigr).
\]
\end{definition}

Ranks add, so $G+K$ is again a game. For morphisms $p\colon G\to G'$ and
$q\colon K\to K'$ the map $(p+q)(g,k)=(p(g),q(k))$ satisfies \eqref{eq:M},
because
$(p+q)[\theta_{G+K}(g,k)]
 =p[\theta_G(g)]\times\{q(k)\}\cup\{p(g)\}\times q[\theta_K(k)]
 =\theta_{G'+K'}(p(g),q(k))$;
thus $+$ is a bifunctor. Its unit \cite[Proposition~4.2]{Hora} is the game
$\unit$ with one position $\star$ and $\theta(\star)=\emptyset$: indeed
$\theta_{\unit+K}(\star,k)=\{\star\}\times\theta_K(k)$, so the projection
$\unit+K\to K$ is an isomorphism of games.

\begin{definition}[{\cite[Definition~B.13]{Hora}}]
For a set $\Lambda$, a \emph{$\Lambda$-labelled hereditarily finite set} is a
pair $w=(T_w,a_w)$ with $T_w$ a finite set of $\Lambda$-labelled hereditarily
finite sets and $a_w\in\Lambda$. The set of all of them is $\HF_\Lambda$.
\end{definition}

\begin{definition}[{\cite[Proposition~B.14]{Hora}}]
The \emph{cofree game} on $\Lambda$ is $R(\Lambda)=(\HF_\Lambda,\theta_R)$ with
$\theta_R(w)=T_w$, equipped with the counit
$\epsilon_\Lambda\colon\HF_\Lambda\to\Lambda$, $\epsilon_\Lambda(w)=a_w$.
\end{definition}

Since children have smaller set-theoretic rank, $R(\Lambda)$ is a game.
Note that $\HF_\Lambda=\bigcup_{n}\HF_{\Lambda,n}$ with
$\HF_{\Lambda,0}=\emptyset$ and $\HF_{\Lambda,n+1}=\Pfin(\HF_{\Lambda,n})\times\Lambda$
is a genuine set, so no size issue arises below.

\begin{lemma}[{cofree transpose; \cite[Proposition~B.14]{Hora}}]\label{lem:cofree}
For every game $K$ and every function $\sigma\colon UK\to\Lambda$ there is a
unique game morphism $\sigma^{\flat}\colon K\to R(\Lambda)$ with
$\epsilon_\Lambda\circ U\sigma^{\flat}=\sigma$. It is given by
$\sigma^{\flat}(k)=\bigl(\{\sigma^{\flat}(k')\mid k'\in\theta_K(k)\},\,\sigma(k)\bigr)$.
\end{lemma}

\begin{proof}
\emph{Existence.} Define $\sigma^{\flat}(k)$ by induction on the rank
$\rho_K(k)$ using the displayed formula; this is legitimate since each
$k'\in\theta_K(k)$ has strictly smaller rank, and the child set is finite
because $\theta_K(k)$ is. By construction
$\theta_R(\sigma^{\flat}(k))=\sigma^{\flat}[\theta_K(k)]$, which is
\eqref{eq:M}, and $\epsilon_\Lambda(\sigma^{\flat}(k))=\sigma(k)$.

\emph{Uniqueness.} Let $h\colon K\to R(\Lambda)$ satisfy
$\epsilon_\Lambda\circ h=\sigma$. Then $h(k)$ has label $\sigma(k)$, and by
\eqref{eq:M} its child set is $h[\theta_K(k)]$. Induction on $\rho_K(k)$
gives $h(k)=\sigma^{\flat}(k)$.
\end{proof}

Finally, a \emph{subgame} of a game $X$ is a subset $B\subseteq UX$ closed
under $\theta_X$, regarded as a game by restricting $\theta_X$; the inclusion
is then a game morphism. We write $\desc(u)$ for the least set containing $u$
and closed under $\theta$.

\begin{lemma}[image factorisation]\label{lem:image}
Let $h\colon K\to X$ be a game morphism and $I=h[UK]$. Then $I$ is a subgame
of $X$, the corestriction $K\to I$ is a game morphism and is surjective on
positions, and $h$ factors as $K\twoheadrightarrow I\hookrightarrow X$.
\end{lemma}

\begin{proof}
For $y=h(k)\in I$ we have $\theta_X(y)=h[\theta_K(k)]\subseteq I$, so $I$ is
closed under options; the same equality is \eqref{eq:M} for the corestriction.
\end{proof}

\section{The maximal compatible subgame}\label{sec:A}

Fix games $G$ and $H$ and put $S=\Set(UG,UH)$. We read the label of a position
$w$ of $R(S)$ as a function $a_w\colon UG\to UH$.

\begin{definition}\label{def:A}
$A(G,H)\subseteq\HF_S$ is the set of $u$ such that \eqref{eq:E} holds for every
$w\in\desc(u)$ and every $g\in UG$.
\end{definition}

\begin{lemma}\label{lem:subgame}
$A(G,H)$ is a subgame of $R(S)$.
\end{lemma}

\begin{proof}
If $u\in A(G,H)$ and $v\in\theta_R(u)$ then $\desc(v)\subseteq\desc(u)$, so
\eqref{eq:E} holds throughout $\desc(v)$; hence $v\in A(G,H)$.
\end{proof}

\begin{proposition}[evaluation]\label{prop:ev}
The function $\ev_{G,H}\colon UG\times A(G,H)\to UH$, $\ev_{G,H}(g,u)=a_u(g)$,
is a game morphism $G+A(G,H)\to H$.
\end{proposition}

\begin{proof}
Using Lemma~\ref{lem:subgame} to identify $\theta_{A}(u)$ with $\theta_R(u)$,
\[
\ev_{G,H}\bigl[\theta_{G+A}(g,u)\bigr]
=\{a_u(g')\mid g'\in\theta_G(g)\}\cup\{a_v(g)\mid v\in\theta_R(u)\}
=\theta_H\bigl(a_u(g)\bigr)
\]
by \eqref{eq:E} at $u$, which is \eqref{eq:M} for $\ev_{G,H}$.
\end{proof}

\begin{proposition}[maximality]\label{prop:max}
Let $B\subseteq\HF_S$ be a subgame of $R(S)$ such that
$(g,u)\mapsto a_u(g)$ is a game morphism $G+B\to H$. Then $B\subseteq A(G,H)$.
\end{proposition}

\begin{proof}
Let $u\in B$. Writing out \eqref{eq:M} for this morphism at $(g,u)$ and using
$\theta_B(u)=\theta_R(u)$ gives exactly \eqref{eq:E} at $u$, for every
$g\in UG$. Since $B$ is closed under options, $\desc(u)\subseteq B$ and the
same holds at every $w\in\desc(u)$. Hence $u\in A(G,H)$.
\end{proof}

Thus $A(G,H)$ admits a description independent of Definition~\ref{def:A}: it
is the largest subgame of $R(\Set(UG,UH))$ on which label evaluation is a game
morphism out of $G+(-)$.

\section{The adjunction}

\begin{theorem}\label{thm:main}
For all games $G,K,H$ the assignment
\[
  \lambda\colon\catGame(G+K,H)\longrightarrow\catGame\bigl(K,A(G,H)\bigr),
  \qquad \lambda f=\bigl(\sigma_f\bigr)^{\flat},\quad
  \sigma_f(k)(g)=f(g,k),
\]
is a well-defined bijection, with inverse
$h\mapsto \ev_{G,H}\circ(\mathrm{id}_G+h)$. Consequently $G+(-)$ is left
adjoint to $A(G,-)$, and $A(G,H)=\ihom{G}{H}$.
\end{theorem}

\begin{proof}
\emph{$\lambda$ lands in $A(G,H)$.} Let $f\colon G+K\to H$. Condition
\eqref{eq:M} for $f$ at $(g,k)$ reads
\stepcounter{equation}\begin{equation}\tag{F}\label{eq:F}
  \theta_H\bigl(f(g,k)\bigr)=\{f(g',k)\mid g'\in\theta_G(g)\}\cup
  \{f(g,k')\mid k'\in\theta_K(k)\}.
\end{equation}
Let $h_f=(\sigma_f)^{\flat}\colon K\to R(S)$ be the transpose of
Lemma~\ref{lem:cofree} and put $u=h_f(k)$. By that lemma $a_u=\sigma_f(k)$,
i.e.\ $a_u(g)=f(g,k)$, and $\theta_R(u)=\{h_f(k')\mid k'\in\theta_K(k)\}$ with
$a_{h_f(k')}(g)=f(g,k')$. Substituting, \eqref{eq:F} \emph{is} \eqref{eq:E}
at $u$. Now let $I=h_f[UK]$; by Lemma~\ref{lem:image} it is a subgame of
$R(S)$, and every element of $I$ is of the form $h_f(k')$, so \eqref{eq:E}
holds at every element of $I$. Hence label evaluation is a game morphism
$G+I\to H$ by the computation of Proposition~\ref{prop:ev}, and
Proposition~\ref{prop:max} yields $I\subseteq A(G,H)$. Therefore $h_f$
corestricts to a game morphism $\lambda f\colon K\to A(G,H)$, uniquely so
because $A(G,H)\hookrightarrow R(S)$ is injective on positions.

\emph{The inverse is well defined.} For $h\colon K\to A(G,H)$ the composite
$\ev_{G,H}\circ(\mathrm{id}_G+h)$ is a game morphism $G+K\to H$ by
Proposition~\ref{prop:ev} and bifunctoriality of $+$; on positions it is
$(g,k)\mapsto a_{h(k)}(g)$.

\emph{Round trip from $f$.} The label of $\lambda f(k)$ is $\sigma_f(k)$, so
$\ev_{G,H}(\mathrm{id}_G+\lambda f)(g,k)=\sigma_f(k)(g)=f(g,k)$.

\emph{Round trip from $h$.} Put $f=\ev_{G,H}\circ(\mathrm{id}_G+h)$, so
$\sigma_f(k)(g)=a_{h(k)}(g)$, i.e.\ $\sigma_f(k)=a_{h(k)}$. Let
$\iota\colon A(G,H)\hookrightarrow R(S)$ be the inclusion. Then
$\iota\circ h\colon K\to R(S)$ is a game morphism whose composite with
$\epsilon_S$ is $k\mapsto a_{h(k)}=\sigma_f$. By the uniqueness in
Lemma~\ref{lem:cofree}, $\iota\circ h=(\sigma_f)^{\flat}=\iota\circ\lambda f$,
and $\iota$ is injective on positions, so $h=\lambda f$.
\end{proof}

\begin{proposition}[functoriality and naturality]\label{prop:nat}
$A(-,-)\colon\catGame^{\mathrm{op}}\times\catGame\to\catGame$ is a functor, and the
bijection of Theorem~\ref{thm:main} is natural in $G$, $K$ and $H$.
\end{proposition}

\begin{proof}
Let $p\colon G'\to G$ and $r\colon H\to H'$, put $S'=\Set(UG',UH')$ and
define $\Phi_{p,r}\colon S\to S'$ by $\Phi_{p,r}(a)=r\circ a\circ p$.
The cofree functor gives $R(\Phi_{p,r})\colon R(S)\to R(S')$ by
$R(\Phi)(T_w,a_w)=(\{R(\Phi)(v)\mid v\in T_w\},\Phi(a_w))$; this satisfies
\eqref{eq:M} by construction, even though distinct children may be identified
when $\Phi$ is not injective, since both sides of \eqref{eq:M} are direct
images.

Let $u\in A(G,H)$. The descendants of $R(\Phi)(u)$ are exactly the
$R(\Phi)(w)$ with $w\in\desc(u)$. For such $w$ and $g'\in UG'$, exactness of
$r$, then \eqref{eq:E} at $w$, then exactness of $p$ give
\begin{align*}
\theta_{H'}\bigl(r(a_w(p(g'))\bigr)
&=r\bigl[\theta_H(a_w(p(g')))\bigr]\\
&=\{r(a_w(g''))\mid g''\in\theta_G(p(g'))\}\cup\{r(a_v(p(g')))\mid v\in\theta_R(w)\}\\
&=\{\Phi(a_w)(g_1)\mid g_1\in\theta_{G'}(g')\}\cup\{\Phi(a_v)(g')\mid v\in\theta_R(w)\},
\end{align*}
which is \eqref{eq:E} for $R(\Phi)(w)$. Hence $R(\Phi_{p,r})$ restricts to
$A(p,r)\colon A(G,H)\to A(G',H')$. Since $\Phi_{\mathrm{id},\mathrm{id}}=\mathrm{id}$
and $\Phi_{p\circ p',\,r'\circ r}=\Phi_{p',r'}\circ\Phi_{p,r}$, functoriality
follows from that of $R$ together with the uniqueness of restrictions along the
injections $A\hookrightarrow R$.

For naturality let also $q\colon K'\to K$ and $f\colon G+K\to H$, and set
$f'=r\circ f\circ(p+q)$. Then
$\sigma_{f'}(k')(g')=r\bigl(f(p(g'),q(k'))\bigr)$. On the other hand
$\iota'\circ A(p,r)\circ\lambda f\circ q$ is a game morphism
$K'\to R(S')$ whose label at $k'$ is
$\Phi_{p,r}(\sigma_f(q(k')))=g'\mapsto r(f(p(g'),q(k')))$, the same function.
By Lemma~\ref{lem:cofree} the two morphisms coincide, whence
\[
  \lambda\bigl(r\circ f\circ(p+q)\bigr)=A(p,r)\circ\lambda(f)\circ q .
\]
Naturality of the inverse follows, or may be checked directly on positions:
$\ev(\mathrm{id}+A(p,r)\circ h\circ q)(g',k')=r\bigl(\ev(\mathrm{id}+h)(p(g'),q(k'))\bigr)$.
\end{proof}

\begin{remark}[size and degenerate cases]\label{rem:size}
$S=\Set(UG,UH)$ is a set and $A(G,H)\subseteq\HF_S$, so no class-sized objects
occur. The extreme cases agree with the universal property. If $UG=\emptyset$
then $S$ is a singleton and \eqref{eq:E} is vacuous, so
$A(G,H)=R(1)=\HF$, the terminal game \cite[Proposition~3.31]{Hora}; and
indeed $G+K$ is then empty and $\catGame(G+K,H)$ is a singleton. If
$UG\neq\emptyset$ and $UH=\emptyset$ then $S=\emptyset$, so
$A(G,H)=\emptyset$, the initial game; and $\catGame(G+K,H)$ is a singleton or
empty according as $UK$ is empty or not, matching $\catGame(K,\emptyset)$.
Note also that $\HF=R(1)$ is not a finite game, so the construction does not
restrict to the full subcategory of games with finite position set;
Theorem~\ref{thm:main} is a statement about all of $\catGame$.
\end{remark}

\section{Reading \texorpdfstring{$\ihom{G}{H}$}{[G,H]} as a game}\label{sec:reading}

Equation~\eqref{eq:E} is an equality of two sets, and splitting it into the two
inclusions gives its meaning.

\begin{lemma}\label{lem:reading}
Let $u\in A(G,H)$ and $w\in\desc(u)$.
\begin{enumerate}
\item[(i)] $a_w\colon UG\to UH$ is a morphism of option graphs: if
$g'\in\theta_G(g)$ then $a_w(g')\in\theta_H(a_w(g))$.
\item[(ii)] For every child $v\in\theta_R(w)$ and every $g\in UG$ we have
$a_v(g)\in\theta_H(a_w(g))$: a move in $\ihom{G}{H}$ is a move in $H$ made
simultaneously and uniformly at every position of $G$.
\item[(iii)] Every option of $a_w(g)$ in $H$ arises in exactly one of these two
ways.
\end{enumerate}
\end{lemma}

\begin{proof}
(i) and (ii) are the inclusion $\supseteq$ of \eqref{eq:E} restricted to each of
the two families; (iii) is the inclusion $\subseteq$.
\end{proof}

So a position of $\ihom{G}{H}$ is a \emph{plan} for translating $G$ into $H$:
a map $a$ of option graphs, not required to be exact, together with the finite
set of plans one may move to. Exactness of $a$ is not required of a single
plan; what is required is that the exactness defect of $a$ at $g$ --- the
options of $a(g)$ not of the form $a(g')$ --- be precisely realised by the
available revisions of the plan. Under currying, $\lambda f(k)$ is the plan
$g\mapsto f(g,k)$, and its available revisions are the plans $g\mapsto f(g,k')$
for the options $k'$ of $k$. Note that a plan with no revisions available is
precisely an exact morphism $G\to H$; more generally,
Lemma~\ref{lem:reading}(iii) says that the moves of $\ihom{G}{H}$ measure
exactly how far each plan is from being a morphism.

\begin{example}[the extra Nim token]\label{ex:star-general}
Take $G=\ast1$, the game with positions $\{1,0\}$, $\theta(1)=\{0\}$,
$\theta(0)=\emptyset$; playing in $\ast1+K$ is playing $K$ with one extra
token to spend. Write the label of a node $w$ as the pair
$(x_w,y_w)=(a_w(1),a_w(0))$. Then \eqref{eq:E} becomes
\[
  \theta_H(y_w)=\{y_v\mid v\in\theta_R(w)\},
  \qquad
  \theta_H(x_w)=\{y_w\}\cup\{x_v\mid v\in\theta_R(w)\}.
\]
The first says the $y$-labelling is an exact morphism into $H$. The second
forces $y_w\in\theta_H(x_w)$: spending the token must be a genuine move
$x_w\to y_w$ of $H$.
\end{example}

\section{Computations}\label{sec:examples}

Throughout, $\ast n$ denotes the one-heap Nim game with $n$ tokens: positions
$\{0,1,\dots,n\}$ with $\theta(i)=\{0,\dots,i-1\}$; thus $\ast0=\unit$. Call a
game \emph{discrete} if $\theta\equiv\emptyset$. All computations below were
also checked by exhaustive machine enumeration of $A(G,H)$ and of both hom-sets
for $K\in\{\emptyset,\unit,\ast1,\ast2\}$ and the two-element discrete game.

\begin{proposition}[unit law]\label{prop:unit}
$\ihom{\unit}{H}\cong H$, naturally in $H$.
\end{proposition}

\begin{proof}
Here $S\cong UH$; write $h_w\in UH$ for the label of $w$. Condition
\eqref{eq:E} reduces to $\theta_H(h_w)=\{h_v\mid v\in\theta_R(w)\}$ at every
descendant, i.e.\ to the statement that $w\mapsto h_w$ is exact. We claim
$w\mapsto h_w$ is a bijection $A(\unit,H)\to UH$. \emph{Injectivity} is by
induction on the set-theoretic rank of $w$: the children of $w$ are compatible
of smaller rank, so by induction each child is determined by its label, and the
set of children's labels is $\theta_H(h_w)$; hence the child set, and so $w$,
is determined by $h_w$. \emph{Surjectivity} is by induction on $\rho_H$:
given $h$, put $\widehat h=(\{\widehat{h'}\mid h'\in\theta_H(h)\},h)$.
The bijection is a game morphism in both directions by construction.
Naturality is Proposition~\ref{prop:nat}. Alternatively, this is forced by
$\unit+K\cong K$ and Theorem~\ref{thm:main}.
\end{proof}

\begin{proposition}\label{prop:Gunit}
$\ihom{G}{\unit}$ is the terminal game $\HF$ if $UG=\emptyset$; the unit
$\unit$ if $UG\ne\emptyset$ and $G$ is discrete; and the initial game
$\emptyset$ otherwise.
\end{proposition}

\begin{proof}
The first case is Remark~\ref{rem:size}. Otherwise $S$ is a singleton, so
$R(S)=\HF$, and \eqref{eq:E} at $w$ for $g\in UG$ reads
$\emptyset=\theta_{\unit}(\star)
 =\{\star\mid \theta_G(g)\ne\emptyset\}\cup\{\star\mid\theta_R(w)\ne\emptyset\}$.
If $G$ is discrete this says exactly $\theta_R(w)=\emptyset$, so
$A(G,\unit)=\{\emptyset\}\cong\unit$. If some $g_0$ has an option, the
condition fails at every $w$, so $A(G,\unit)=\emptyset$.
\end{proof}

The second case is easy to confirm against the universal property: a morphism
$X\to\unit$ exists (and is then unique) iff $X$ is discrete, and $G+K$ is
discrete iff both $G$ and $K$ are.

\begin{proposition}\label{prop:nimbers}
$\ihom{\ast1}{\ast n}\cong\unit$ for every $n\ge1$, and
$\ihom{\ast1}{\ast0}=\ihom{\ast2}{\ast1}=\emptyset$.
\end{proposition}

\begin{proof}
For $\ihom{\ast1}{\ast n}$ use the two equations of
Example~\ref{ex:star-general} with $H=\ast n$.

If $y_w=0$ then $\theta_H(y_w)=\emptyset$, so $w$ has no children, and the
second equation gives $\theta_H(x_w)=\{0\}$, forcing $x_w=1$. So the only
node with $y_w=0$ is the leaf $\ell$ with label $(1,0)$.

If $y_w=1$ then all children have $y$-label $0$, hence all children equal
$\ell$; as children form a set, $w$ has exactly one child, namely $\ell$, whose
$x$-label is $1$. The second equation gives $\theta_H(x_w)=\{1\}\cup\{1\}=\{1\}$.
But in $\ast n$ every non-terminal position has $0$ among its options and the
terminal position has none, so no position has option set $\{1\}$:
contradiction. Hence no node has $y_w=1$.

If $y_w=j\ge2$ then the children's $y$-labels are $\{0,\dots,j-1\}\ni1$, so
some child has $y$-label $1$: impossible by the previous paragraph.

Therefore $A(\ast1,\ast n)=\{\ell\}\cong\unit$.

$\ihom{\ast1}{\ast0}=\emptyset$ is Proposition~\ref{prop:Gunit}. For
$\ihom{\ast2}{\ast1}$ write $a_w=(a_w(2),a_w(1),a_w(0))$ with values in
$\{1,0\}$. Equation \eqref{eq:E} at $g=1$ reads
$\theta_H(a_w(1))=\{a_w(0)\}\cup\{a_v(1)\mid v\in\theta_R(w)\}$; the right-hand
side is non-empty, so $a_w(1)=1$ and $\theta_H(1)=\{0\}$, whence $a_w(0)=0$ and
$a_v(1)=0$ for all children $v$. But we have just shown $a_v(1)=1$ for
\emph{every} node $v$; so $w$ has no children. Then \eqref{eq:E} at $g=2$
reads $\theta_H(a_w(2))=\{a_w(1),a_w(0)\}=\{1,0\}$, and no position of $\ast1$
has option set $\{0,1\}$. So no node is compatible.
\end{proof}

\begin{remark}\label{rem:nim}
In Joyal's compact closed category of games and winning strategies
\cite{Joyal}, and in its coalgebraic reworking \cite{HLR}, the object
classifying maps out of $G+(-)$ is $-G+H$, which for impartial games is
$G+H$; for one-heap Nim this is $\ast n+\ast m$, equivalent to
$\ast(n\oplus m)$ by the Sprague--Grundy theorem \cite[Theorem~2.17]{Hora}.
Proposition~\ref{prop:nimbers} gives $\ihom{\ast1}{\ast2}\cong\unit$ where the
Nim-arithmetic answer would be $\ast3$, and
$\ihom{\ast2}{\ast1}=\emptyset$ where it would be $\ast3$ again. The internal
hom of Theorem~\ref{thm:main} is therefore \emph{not} a categorification of
Nim arithmetic, and the collapse is not an artefact of small examples: it comes
from the exactness \eqref{eq:M} of morphisms in $\catGame$, which is far more
rigid than the existence of a winning strategy. It is worth noting that
$\ihom{-}{-}$ can nevertheless be large and non-degenerate --- by
Proposition~\ref{prop:unit} it is $H$ when $G=\unit$, and by
Remark~\ref{rem:size} it is the terminal game when $G$ is empty.
\end{remark}

\begin{example}\label{ex:six}
A non-degenerate case with $G=\ast1$. Let $H$ have positions
$\{d,c,b,e\}$ with $\theta(d)=\{c,b\}$, $\theta(c)=\theta(b)=\{e\}$,
$\theta(e)=\emptyset$. Using Example~\ref{ex:star-general}: the nodes with
$y=e$ are the two leaves $\ell_c=(c,e)$ and $\ell_b=(b,e)$ (a leaf needs
$\theta_H(x)=\{y\}$, and both $c$ and $b$ qualify). A node with $y=c$ has
children among the leaves with $\theta_H(x_w)=\{c\}\cup\{x\text{-labels}\}$;
since $\theta_H(d)=\{c,b\}$ and no position has option set $\{c\}$, the child
set must contain $\ell_b$, giving two nodes $(\,(d,c),\{\ell_b\}\,)$ and
$(\,(d,c),\{\ell_c,\ell_b\}\,)$. Symmetrically $y=b$ gives
$(\,(d,b),\{\ell_c\}\,)$ and $(\,(d,b),\{\ell_c,\ell_b\}\,)$. A node with
$y=d$ would need $\theta_H(x_w)=\{d\}$, impossible. So $\ihom{\ast1}{H}$ has
six positions, two of them terminal, and the four non-terminal ones have one or
two options. The example shows that a position of $\ihom{G}{H}$ genuinely
remembers more than its label: $(\,(d,c),\{\ell_b\}\,)$ and
$(\,(d,c),\{\ell_c,\ell_b\}\,)$ carry the same plan $(d,c)$ but differ in
which revisions are on offer.
\end{example}

\section{Related work}\label{sec:related}

The question answered here is \cite[Question~5.7]{Hora}, restated as
Problem~5.0.5 of \cite{HoraProblems}; that paper supplies the category
$\catGame$, the Conway addition, the abstract closedness
\cite[Proposition~4.2, Corollary~B.11]{Hora} and the cofree game
\cite[Proposition~B.14]{Hora} on which our construction rests, but no explicit
internal hom.

The classical closed structure on games is of a different nature. Joyal
\cite{Joyal} takes the opposite $-G$ of a partisan game, exchanging the roles
of Left and Right, defines a morphism $G\to H$ to be a winning strategy in
$H-G$, and obtains a compact closed category in which the internal hom is
$-G+H$. Honsell, Lenisa and Redamalla \cite{HLR} redo this coalgebraically,
allowing non-well-founded games as elements of a final coalgebra and defining
sum, negation and linear implication by final morphisms; their well-founded
full subcategory recovers Joyal's. In both cases the objects are single games
and the morphisms are strategies. In $\catGame$ the objects are whole position
sets with an option map and the morphisms are exact coalgebra homomorphisms,
so neither the negation nor the copycat strategy is available, and
Remark~\ref{rem:nim} shows the two internal homs genuinely disagree.

Kapulkin and Kershaw \cite{KK} classify the closed symmetric monoidal
structures on the category of graphs and give explicit internal homs for the
box product. That is a useful methodological comparison --- an internal hom
described by combinatorial data on vertices and edges --- but their morphisms
are graph morphisms rather than exact ones, so the formulas do not transfer.
Altman and Lipparini \cite{AL} construct a ring structure on equivalence
classes of partizan games; this concerns algebra on a quotient, not a right
adjoint in a category of games, and is a different problem. Finally, Anel and
Joyal \cite{AJ} carve mapping objects out of cofree coalgebras in the setting
of Sweedler theory; the general shape of the technique --- impose a
compatibility condition inside a cofree object --- is the same as ours, and we
record this as the nearest general antecedent we are aware of, without
claiming a formal implication in either direction.

\section{Open questions}\label{sec:open}

\begin{enumerate}
\item \textbf{Player switching.} \cite{Hora} remarks after Question~5.7 that
the closed structure ``might be better understood when we consider the
partisan games $\mathbf{RecCoalg}_{\Pfin\times\Pfin}$'', and Problem~5.0.2 of
\cite{HoraProblems} asks for a double category whose vertical part is $\catGame$
and whose horizontal part is Joyal's category of games and strategies. Nothing
in this note bears on that. Our $\ihom{G}{H}$ is a vertical internal hom, and
Remark~\ref{rem:nim} shows it is not $-G+H$; we do not know whether it arises
as some vertical shadow of the horizontal structure, nor whether the plans of
Lemma~\ref{lem:reading} admit a strategic reading.

\item \textbf{Other liftings.} The proof of Theorem~\ref{thm:main} uses the
Conway option formula directly, through \eqref{eq:F}. Does the maximal
compatible subgame construction work for an arbitrary bifunctor lifting the
cartesian product of $\Set$, e.g.\ the selective and conjunctive sums
\cite[Examples 4.3, 4.4]{Hora}? This would bear on
\cite[Questions~5.5, 5.6]{Hora}. One would need to redo the closure,
maximality and naturality arguments; we have not done so.

\item \textbf{When is $\ihom{G}{H}$ non-degenerate?}
Proposition~\ref{prop:nimbers} and Example~\ref{ex:six} suggest that
$\ihom{G}{H}$ is empty or trivial unless $H$ is ``wide'' relative to $G$ ---
in Example~\ref{ex:six} what made the difference was that $H$ contained two
distinct positions with the same option set. Is there a clean criterion on
$G,H$ for $\ihom{G}{H}\ne\emptyset$? Equivalently, when does there exist an
exact morphism $G\to H$ after adding enough spare moves?

\item \textbf{Reduction of positions.} A position of $\ihom{G}{H}$ is a
labelled tree, which is a large amount of data; Example~\ref{ex:six} shows
the tree is not determined by its root label. Is there a canonical quotient or
normal form --- say, a bound on the depth that matters, or a presentation by a
subset of $UH^{UG}$ with extra structure?

\item \textbf{Novelty.} We have not located this construction in the
literature (see Section~\ref{sec:related}), but we have not carried out a
systematic search of the coalgebraic mapping-object literature and do not
claim priority.
\end{enumerate}

\begin{thebibliography}{HLR12}

\bibitem[AL26]{AL}
H.~Altman and P.~Lipparini,
\emph{A ring structure on the class of combinatorial games},
arXiv:2604.27847, 2026.
\href{https://arxiv.org/abs/2604.27847}{arXiv:2604.27847}.

\bibitem[AJ13]{AJ}
M.~Anel and A.~Joyal,
\emph{Sweedler theory for (co)algebras and the bar-cobar constructions},
arXiv:1309.6952, 2013.
\href{https://arxiv.org/abs/1309.6952}{arXiv:1309.6952}.

\bibitem[HLR12]{HLR}
F.~Honsell, M.~Lenisa and R.~Redamalla,
\emph{Categories of coalgebraic games},
MFCS 2012, Lecture Notes in Comput. Sci. 7464, Springer, 2012, pp.~503--515.

\bibitem[Hor25]{Hora}
R.~Hora,
\emph{Games as recursive coalgebras: a categorical view on the Nim-sum},
arXiv:2510.22886v3, 2025.
\href{https://arxiv.org/abs/2510.22886}{arXiv:2510.22886}.

\bibitem[HorP]{HoraProblems}
R.~Hora, \emph{Problems},
\href{https://sites.google.com/view/ryuya-hora/problems}{sites.google.com/view/ryuya-hora/problems},
accessed 26 August 2026.

\bibitem[Joy77]{Joyal}
A.~Joyal,
\emph{Remarques sur la th\'eorie des jeux \`a deux personnes},
Gazette des sciences math\'ematiques du Qu\'ebec \textbf{1}(4) (1977),
46--52; English translation by R.~Houston.

\bibitem[KK24]{KK}
K.~Kapulkin and N.~Kershaw,
\emph{Closed symmetric monoidal structures on the category of graphs},
Theory Appl. Categ. \textbf{41}(23) (2024), 760--784.
\href{https://arxiv.org/abs/2310.00493}{arXiv:2310.00493}.

\end{thebibliography}

\end{document}